Gluing Formulas for Volume Forms on Representation Varieties of Surfaces

Publication date: 22 February 2022

Abstract: Let

S i g m a_{g, n}

be a compact oriented surface with genus

g g e q 2

bordered by

n

circles. Due to Witten, the twisted Reidemeister torsion coincides with a power of the Atiyah-Bott-Goldman-Narasimhan symplectic form on the space of representations of

p i_{1} (S i g m a_{g, 0})

in any semi-simple Lie group. In the present paper, we first obtain a multiplicative gluing formula for the twisted Reidemeister torsion of

S i g m a_{g, 0}

in terms of torsions of

S i g m a_{2, 2},

S i g m a_{2, 1},

and boundary circles

m a t h b b S^{1} .

Then, by using Heusener and Porti's results on

S i g m a_{g, n},

we show that the symplectic volume form on the representation variety of

S i g m a_{g, 0}

can be expressed as a product of the holomorphic symplectic volume forms on the relative representation varieties of surfaces

S i g m a_{2, 1}

and

S i g m a_{2, 2} .

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